Sergei Soloviev
IRIT, Toulouse

On Varieties of Symmetric Monoidal Closed Categories

In our talk we consider extensions of the equational theory of SMCC. A
SMCC is a category with additional structure that consists of internal
hom-functor, symmetric tensor product (left adjoint to hom),
distinguished object (tensor unit) and related natural transformations
subject to standard equations such as Mac Lane pentagon and hexagon
(that form its equational theory). It is known that in related case of
Cartesian Closed Categories (with cartesian product instead of tensor)
the equational theory is maximal (there is no non-trivial
extansions). We show that in case of SMCC the situation is
different. We built an infinite series of diagrams {D_n} of canonical
natural transformations with the same graph of naturality conditions
and the models K_k such that in the category K_k the diagrams D_1,...,
D_k are non commutative while D_{k+1},... are commutative. Thus, there
exists infinitely many different varieties (in the sense of universal
algebra) obtained by extension of the equational theory by D_{k+1} (as
a new axiom).  The same is true for weaker theories of closed
categories.  The models K_k are categories of semimodules of certain
type over certain commutative semirings. We discuss also the problem
of dependency of diagrams.